I'm not a big gamer, but I've read lots of things that rail against QTEs, which seem to be cutscenes with some button pressing involved? They seem to be viewed as lazy and as something that detracts from the game, but I'm still not entirely sure what they are. I can't picture a good example in my head. Thanks!

A quick-time event is when a specific button combination flashes onscreen, and you must quickly press it (usually within 1 or 2 seconds) to avoid your character dying to some sudden event.

If you have ever played Resident Evil 4, that was a game that was known for the many quick-time events featured in cutscenes. Here is a video featuring some of them. The first QTE occurs at 0:12.

They are sometimes disliked because they don't feel much like regular gameplay (you don't have control of your character or get to make any decisions, just press the buttons it tells you or game over), but are most disliked when featured in cutscenes, because they mean you can't just sit back and watch the scene like a movie, and that you must re-watch the scene again and again each time you fail. On hard mode you might watch the same 5-minute scene 10 times before beating it which is very tedious.

In the video, the only thing the player is doing is mashing whatever button appears on screen. First he presses circle repeatedly (which loads the catapult), then square to dodge correctly, etc. You fail if you don't hit the right button in time, or hit the wrong button.

They're very common in videogames nowadays, and are generally extremely easy and, to some, a sign of poor game design. Personally I don't have a problem with them as long as they are used sparingly; they typically are just there to make cutscenes more fun and interactive.